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Let P,M,N be n$\times$n matrices such that M and N are non singular.If x is an eigenvector of P corresponding to eigen value $\lambda$,then an eigenvector of N$^{-1}M$PM$^{-1}N$ corresponding to eigenvalue $\lambda$is

(a) MN$^{-1}$x (b) M$^{-1}Nx$ (c) NM$^{-1}x$ (d) N$^{-1}Mx$

One more thing that worries me is P and N$^{-1}MPM^{-1}N$ having same eigen value .What makes it necessary?

My Approach : The only thing i know is since M and N are non-singular N$^{-1}M$ and M$^{-1}N$ they will have same set of eigenvalues.I don't know if it has anything to do with question.

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As already said $Px=\lambda x$.

$$N^{-1}MPM^{-1}N=K \rightarrow N^{-1}MP=KN^{-1}M \rightarrow N^{-1}MPx=K(N^{-1}Mx) \rightarrow \lambda (N^{-1}Mx)=K(N^{-1}Mx) $$ and so, $K$ has eigenvalue $\lambda$ and eigenvector $N^{-1}Mx$

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Hint: let $Px=\lambda x$ and $B\equiv APA^{-1}$ for some non-singular matrix $A$. Then $$ B\color{red}{Ax}=APA^{-1}Ax=\lambda \color{red}{Ax} $$

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